
Find the point of intersection of normals at and on the parabola .
Answer
531.6k+ views
Hint: The given problem is related to the equation of normal to parabola in parametric form. The general equation of the normal to the parabola at a point is given by , where is a parameter.
Complete step-by-step answer:
We are given the equation of the parabola as .
Now, we will consider two points on the parabola given by and , where , and are parameters.
Now, we need to find the equation of normal at and .
We know, the general equation of the normal to the parabola at a point is given by , where is a parameter.
So, the normal to the parabola at the point will be given is given by substituting in place of in the general equation of the normal.
On substituting in place of in the general equation of the normal, we get .
And the normal to the parabola at the point is given as .
Now, to find the point of intersection of and , we substitute the value of from equation in equation .
On substituting the value of from equation in equation , we get .
Now, we will substitute in equation .
On substituting in equation , we get .
So, the point of intersection of the normal is .
Note: While simplifying the equations, please make sure that sign mistakes do not occur. These mistakes are very common and can confuse while solving. Ultimately the answer becomes wrong. So, sign conventions should be carefully taken.
Complete step-by-step answer:
We are given the equation of the parabola as
Now, we will consider two points on the parabola given by

Now, we need to find the equation of normal at
We know, the general equation of the normal to the parabola at a point
So, the normal to the parabola at the point
On substituting
And the normal to the parabola at the point
Now, to find the point of intersection of
On substituting the value of
Now, we will substitute
On substituting
So, the point of intersection of the normal is
Note: While simplifying the equations, please make sure that sign mistakes do not occur. These mistakes are very common and can confuse while solving. Ultimately the answer becomes wrong. So, sign conventions should be carefully taken.
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