
The number of points with integral coordinates \[(2a,a-1)\]that fall in the interior
of the larger segment of the circle \[{{x}^{2}}+{{y}^{2}}=25\] cut off by the parabola
x\[{{x}^{2}}+4y=0\]is
(a) One
(b) Two
(c) Three
(d) None of these
Answer
513.9k+ views
Hint: We will substitute the coordinates of points in the equation of both curves
and check if the value is positive or negative.
We have a circle\[{{x}^{2}}+{{y}^{2}}=25\] and a parabola\[{{x}^{2}}+4y=0\].
We want to find the points with integral coordinates that fall in the interior of the larger
segment of the circle cut off by the parabola.
To find the points which lie in the larger segment of the circle \[{{x}^{2}}+{{y}^{2}}=25\]cut off
by the parabola\[{{x}^{2}}+4y=0\], we will substitute the points in the equation of the circle
and the parabola.
Substituting the point \[(2a,a-1)\]in the equation of parabola\[{{x}^{2}}+4y=0\], we will get
positive value as the above equation has no real roots.
Substituting the values, we get
\[\begin{align}
& \Rightarrow {{(2a)}^{2}}+4(a-1)>0 \\
& \Rightarrow 4{{a}^{2}}+4a-4>0 \\
& \Rightarrow {{a}^{2}}+a-1>0 \\
\end{align}\]
We observe that this holds true for all\[a>1\].
While, substituting the point \[(2a,a-1)\]in the equation of circle\[{{x}^{2}}+{{y}^{2}}=25\], we
will get a negative value.
Substituting the values, we get
\[\begin{align}
& \Rightarrow {{(2a)}^{2}}+{{(a-1)}^{2}}-25<0 \\
& \Rightarrow 5{{a}^{2}}-2a-24<0 \\
\end{align}\]
We will now factorize the above equation.
\[\begin{align}
& \Rightarrow 5{{a}^{2}}+10a-12a-24<0 \\
& \Rightarrow 5a(a+2)-12(a+2)<0 \\
\end{align}\]
So, we have \[(a+2)(5a-12)<0\]
If\[a+2>0,5a-12<0\], the possible values of\[a\]are {-1, 0, 1, 2}.
If\[a+2<0,5a-12>0\], no such value of \[a\]exists.
As 4 different values of \[a\]exist, we have 4 different points that lie in the larger segment of
the circle bounded by the parabola.
Hence, the correct answer is None of these
Note: It is very necessary to keep in mind that we want to find the points lying in the larger
section of the circle bounded by the parabola and not the smaller segment. if we solve the
question for the smaller segment, we will get equations with opposite inequalities and thus,
we will get a wrong answer.
and check if the value is positive or negative.
We have a circle\[{{x}^{2}}+{{y}^{2}}=25\] and a parabola\[{{x}^{2}}+4y=0\].
We want to find the points with integral coordinates that fall in the interior of the larger
segment of the circle cut off by the parabola.
To find the points which lie in the larger segment of the circle \[{{x}^{2}}+{{y}^{2}}=25\]cut off
by the parabola\[{{x}^{2}}+4y=0\], we will substitute the points in the equation of the circle
and the parabola.

Substituting the point \[(2a,a-1)\]in the equation of parabola\[{{x}^{2}}+4y=0\], we will get
positive value as the above equation has no real roots.
Substituting the values, we get
\[\begin{align}
& \Rightarrow {{(2a)}^{2}}+4(a-1)>0 \\
& \Rightarrow 4{{a}^{2}}+4a-4>0 \\
& \Rightarrow {{a}^{2}}+a-1>0 \\
\end{align}\]
We observe that this holds true for all\[a>1\].
While, substituting the point \[(2a,a-1)\]in the equation of circle\[{{x}^{2}}+{{y}^{2}}=25\], we
will get a negative value.
Substituting the values, we get
\[\begin{align}
& \Rightarrow {{(2a)}^{2}}+{{(a-1)}^{2}}-25<0 \\
& \Rightarrow 5{{a}^{2}}-2a-24<0 \\
\end{align}\]
We will now factorize the above equation.
\[\begin{align}
& \Rightarrow 5{{a}^{2}}+10a-12a-24<0 \\
& \Rightarrow 5a(a+2)-12(a+2)<0 \\
\end{align}\]
So, we have \[(a+2)(5a-12)<0\]
If\[a+2>0,5a-12<0\], the possible values of\[a\]are {-1, 0, 1, 2}.
If\[a+2<0,5a-12>0\], no such value of \[a\]exists.
As 4 different values of \[a\]exist, we have 4 different points that lie in the larger segment of
the circle bounded by the parabola.
Hence, the correct answer is None of these
Note: It is very necessary to keep in mind that we want to find the points lying in the larger
section of the circle bounded by the parabola and not the smaller segment. if we solve the
question for the smaller segment, we will get equations with opposite inequalities and thus,
we will get a wrong answer.
Recently Updated Pages
Master Class 11 Accountancy: Engaging Questions & Answers for Success

Glucose when reduced with HI and red Phosphorus gives class 11 chemistry CBSE

The highest possible oxidation states of Uranium and class 11 chemistry CBSE

Find the value of x if the mode of the following data class 11 maths CBSE

Which of the following can be used in the Friedel Crafts class 11 chemistry CBSE

A sphere of mass 40 kg is attracted by a second sphere class 11 physics CBSE

Trending doubts
10 examples of friction in our daily life

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Difference Between Prokaryotic Cells and Eukaryotic Cells

State and prove Bernoullis theorem class 11 physics CBSE

What organs are located on the left side of your body class 11 biology CBSE

Write down 5 differences between Ntype and Ptype s class 11 physics CBSE
