Answer
Verified
425.4k+ views
Hint: To find the locus of point of intersection of perpendicular tangents to the parabola, one must know that the point of intersection of perpendicular tangents to any curve is the equation of the director circle of the curve which is the directrix in the case of a parabola.
Complete step-by-step answer:
We have a curve \[{{y}^{2}}+4y-6x-2=0\].
We want to find the locus of points from which perpendicular tangents can be drawn to the parabola.
We know that the locus of points of intersection of the perpendicular tangents to a curve is called the director circle.
Rewriting the equation of the curve, we get \[{{y}^{2}}+4y=6x+2\]
Adding 4 on both sides of the equation, we get \[{{y}^{2}}+4y+4=6x+6\]
\[\Rightarrow {{\left( y+2 \right)}^{2}}=4\times \dfrac{3}{2}\left( x+1 \right)\]
This is the equation of parabola of a curve. We know for a parabola, the equation of the director circle is the equation of the directrix of parabola.
We know that the directrix is a straight line and not a circle, but it can be assumed to be a circle of infinite radius.
We know the equation of the directrix of parabola of the form \[{{\left( y-k \right)}^{2}}=4a\left( x-h \right)\] is \[x+a-h=0\].
Comparing with the equation of our parabola, we get \[k=-2,a=\dfrac{3}{2},h=-1\]
Thus, the equation of director circle (directrix) is \[x+\dfrac{3}{2}+1=0\]
\[\Rightarrow 2x+5=0\], which is the required equation of the director circle.
However, it is not necessary that we will always get the locus of intersection of tangents as a straight line. If the angle between the two tangents changes, the equation of locus of their point of intersection will also change depending upon the angle between the two tangents.
Hence, the correct answer is d) \[2x+5=0\].
Note: There’s no need to find the exact equations of tangent and solve for locus. It will lead to a time consuming and longer solution.
Complete step-by-step answer:
We have a curve \[{{y}^{2}}+4y-6x-2=0\].
We want to find the locus of points from which perpendicular tangents can be drawn to the parabola.
We know that the locus of points of intersection of the perpendicular tangents to a curve is called the director circle.
Rewriting the equation of the curve, we get \[{{y}^{2}}+4y=6x+2\]
Adding 4 on both sides of the equation, we get \[{{y}^{2}}+4y+4=6x+6\]
\[\Rightarrow {{\left( y+2 \right)}^{2}}=4\times \dfrac{3}{2}\left( x+1 \right)\]
This is the equation of parabola of a curve. We know for a parabola, the equation of the director circle is the equation of the directrix of parabola.
We know that the directrix is a straight line and not a circle, but it can be assumed to be a circle of infinite radius.
We know the equation of the directrix of parabola of the form \[{{\left( y-k \right)}^{2}}=4a\left( x-h \right)\] is \[x+a-h=0\].
Comparing with the equation of our parabola, we get \[k=-2,a=\dfrac{3}{2},h=-1\]
Thus, the equation of director circle (directrix) is \[x+\dfrac{3}{2}+1=0\]
\[\Rightarrow 2x+5=0\], which is the required equation of the director circle.
However, it is not necessary that we will always get the locus of intersection of tangents as a straight line. If the angle between the two tangents changes, the equation of locus of their point of intersection will also change depending upon the angle between the two tangents.
Hence, the correct answer is d) \[2x+5=0\].
Note: There’s no need to find the exact equations of tangent and solve for locus. It will lead to a time consuming and longer solution.
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Summary of the poem Where the Mind is Without Fear class 8 english CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Write an application to the principal requesting five class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE