Question

Consider the parabola $(x-1{)}^2+(y-2{)}^2=\frac{(12x-5y+3{)}^2}{169}$
$\begin{array}{cc}ColumnI& ColumnII\\ \text{a. Locus of point of intersection of perpendicular tangent}& \text{p.}(12x-5y-2=0)\\ \text{b. Locus of foot of perpendicular from focus upon any tangent}& \text{q.}(5x+12y-29=0)\\ \text{c. Line along which minimum length of focal chord occurs}& \text{r.}(12x-5y+3=0)\\ \text{d. Line about which parabola is symmetrical}& \text{s.}(24x-10y+1=0)\end{array}$

• A. $\text{a}-\text{r},\text{b}-\text{s},\text{c}-\text{p},\text{d}-\text{q}$
• B. $\text{a}-\text{s},\text{b}-\text{r},\text{c}-\text{p},\text{d}-\text{q}$
• C. $\text{a}-\text{r},\text{b}-\text{q},\text{c}-\text{p},\text{d}-\text{s}$
• D. $\text{a}-\text{p},\text{b}-\text{s},\text{c}-\text{q},\text{d}-\text{r}$

Answer 1

The correct option is A $\text{a}-\text{r},\text{b}-\text{s},\text{c}-\text{p},\text{d}-\text{q}$

The locus of the point of intersection of perpendicular tangent is directrix, which is $12x-5y+3=0.$

The parabola is symmetrical about its axis, which is a line passing through the focus (1, 2) and perpendicular to the directirx, which has equaiton $5x+12y-29=0.$

The minimum length of focal chord occurs along the latus rectum line, which is a line passing through the focus and parallel to the directirx, i.e., $12x-5y-2=0.$

The locus of the foot of perpendicular from the focus upon any tangent is tangent at the vertex, which is parallel to the directrix and equidistant from the directrix and latus rectum line, i.e., $12-5y+\lambda =0$
where $\frac{|\lambda -3|}{\sqrt{{12}^2+5^2}}=\frac{|\lambda +2|}{\sqrt{{12}^2+5^2}}or\lambda =\frac{1}{2}$
Hence, the equation of tangent at vertex is $24x-10y+1=0.$