7.2 Parametric form

  Ellipse

Equation of tangent at a point if the coordinates of point is in parametric from i.e., $(a\cos \theta ,b\sin \theta )$ is $$\frac{{x\cos \theta }}{a} + \frac{{y\sin \theta }}{b} = 1$$

Question 8. A tangent is drawn on any point on the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ which cuts the coordinate axis at $A$ and $B$. Find the locus of mid-point of a line $AB$.

Solution: Let us assume the mid-point of line $AB$ be $M(h,k)$ and $P(a\cos \theta ,b\sin \theta )$ be the point on the ellipse at which a tangent is drawn as shown in figure $14$.

The equation of tangent at $P$ using parametric form is $$\frac{{x\cos \theta }}{a} + \frac{{y\sin \theta }}{b} = 1...(1)$$

Therefore, the coordinates of point $A$ can be find out by putting $x=0$ in equation $(1)$ i.e., $A(0,\frac{b}{{\sin \theta }})$ and coordinates of point $B$ can be find out by putting $y=0$ in equation $(1)$ i.e., $B(\frac{a}{{\cos \theta }},0)$.

Now, using mid-point fomulae, we get $$h = \frac{a}{{2\cos \theta }}$$ and $$k = \frac{b}{{2\sin \theta }}$$

We can also write it as, $\cos \theta = \frac{a}{{2h}}...(2)$ and $\sin \theta = \frac{b}{{2k}}...(3)$.

Squaring and adding equations $(2)$ and $(3)$, we get

$$1 = \frac{{{a^2}}}{{4{h^2}}} + \frac{{{b^2}}}{{4{k^2}}}$$

Therefore, the locus of mid-point of line $AB$ is $$4 = \frac{{{a^2}}}{{{x^2}}} + \frac{{{b^2}}}{{{y^2}}}$$